The paper is devoted to the numerical behaviour of the \(s\)-step Lanczos algorithm and the \(s\)-step variant of the conjugate gradient (CG) method. These schemes have a potential to improve the parallel performance of the classical formulations of Lanczos and CG that are often limited by communication. However, there is a trade-off between potential parallelism and numerical stability of these schemes. Indeed, the reduction of the synchronization cost per iteration comes with a price that the \(s\)-step formulations behave quite differently in finite precision arithmetic and potentially lead to faster loss of orthogonality and slower convergence rate when compared to classical but more sequential algorithms. In this work the authors consider the multi-precision environment that has now become a crucial part of supercomputer parallel architectures. They investigate whether the use of multiple precision in the computation can improve the numerical behaviour of \(s\)-step Lanczos and CG algorithms without a significant performance overhead. This is achieved by doubling the working accuracy in computation of entries and in matrix-vector multiplication by the associated Gram matrices at each outer loop iteration, while all other computations are performed with the standard precision. The rounding error analysis in the paper leads to an improvement of existing bounds; the error terms depend here only linearly on the conditioning of the generated \(s\)-step bases. Numerical experiments also demonstrate that mixed precision approach leads to the improved numerical behaviour for both \(s\)-step Lanczos and CG algorithms.